5
5.0

Dec 30, 2019
12/19

Dec 30, 2019
by
Jan de Leeuw

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The paper recasts the minimization of the Gifi loss function as a pair of nonlinear eigenvalue problems of optimally scaled data (or a nonlinear singular value problem of optimally scaled data). The first and second derivatives of the eigenvalues with respect to the data transformations are computed.

Topics: Multivariate Analysis, R Programming

5
5.0

Nov 11, 2019
11/19

Nov 11, 2019
by
Jan de Leeuw

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Gifi's Eigenvalues

Topics: Multivariate Analysis, Gifi, Eigenvalues, R Programming

3
3.0

Oct 23, 2019
10/19

Oct 23, 2019
by
Jan de Leeuw

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These are notes to explain the update of the Gifi programs for nonlinear multivariate analysis, currently in progress.

Topics: Multivariate Analysis, Gifi, R Programming

10
10.0

Aug 27, 2019
08/19

Aug 27, 2019
by
Jan de Leeuw

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To study convergence of SMACOF we introduce a modiﬁcation mSMACOF that rotates the conﬁgurations from each of the SMACOF iterations to principal components. This modiﬁcation, called mSMACOF, has the same stress values as SMACOF in each iteration, but unlike SMACOF it produces a sequence of conﬁgurations that properly converges to a solution. We show that the modiﬁed algorithm can be implemented by iterating ordinary SMACOF to convergence, and then rotating the SMACOF solution to...

Topics: Multidimensional Scaling, SMACOF, Majorization, MM Algorithm, R Programming

11
11

Aug 21, 2019
08/19

Aug 21, 2019
by
Jan de Leeuw

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We review the continuity and diﬀerentiability properties of the stress loss function in MDS and the local and global convergence properties of the SMACOF algorithm.

Topics: Multidimensional Scaling, Stress, SMACOF, R Programming, Majorization, MM Algorithm

15
15

Jun 19, 2019
06/19

Jun 19, 2019
by
Jan de Leeuw

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If the n × p matrix X is a stationary point of the MDS loss function, then it is also the global minimum over the subspace of all n × p matrices with the same column space as X.

Topics: Multidimensional Scaling, Full-dimensional Scaling, SMACOF

5
5.0

Jun 9, 2019
06/19

Jun 9, 2019
by
Jan de Leeuw

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Majorization theory for families of majorizers is developed in the one-dimensional case. This generalizes the notion of a sharp quadratic majorizer discussed by De Leeuw and Lange (2009)

Topics: Optimization, Majorization, MM Algorithm

18
18

Jun 7, 2019
06/19

Jun 7, 2019
by
Jan de Leeuw

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The full-dimensional (metric, Euclidean, least squares) multidimensional scaling stress loss function is combined with a quadratic external penalty function term. The trajectory of minimizers of stress for increasing values of the penalty parameter is then used to ﬁnd (tentative) global minima for low-dimensional multidimensional scaling. This is illustrated with several one-dimensional and two-dimensional examples.

Topics: Multidimensional Scaling, Global Optima, Full-dimensional Scaling, SMACOF, Exterior Penalty...

12
12

May 9, 2019
05/19

May 9, 2019
by
Jan de Leeuw

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This describes a C version of a infeasible primal-dual algorithm for positive deﬁnite quadratic programming with box constraints, proposed by Voglis and Lagaris. We also give a straightforward .C() interface for R.

Topics: Optimization, Quadratic Programming, R Programming

8
8.0

May 3, 2019
05/19

May 3, 2019
by
Jan de Leeuw

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We compute pseudo-conﬁdence ellipses around MDS solutions, using a new fast implementation of the Hessian of the stress loss function.

Topics: Multidimensional Scaling, R Programming, Stability, Confidence Regions

7
7.0

Apr 29, 2019
04/19

Apr 29, 2019
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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In multidimensional scaling we map dissimilarity matrices into configurations for which stress is stationary. In inverse MDS we map configurations into dissimilarity matrices for which stress is stationary.

Topics: Multidimensional Scaling, SMACOF, Inverse Multidimensional Scaling, R Programming

13
13

Apr 29, 2019
04/19

Apr 29, 2019
by
Jan de Leeuw

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We discuss projection on the intersection of a polyhedral convex cone and a sphere, in particular when the target is in the polar cone. We also discuss projection on the double cone formed by the cone and its negative.

Topics: Optimization, Regression, Projection

11
11

Apr 24, 2019
04/19

Apr 24, 2019
by
Jan de Leeuw

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The stress loss function in metric multidimensional scaling is diﬀerentiable at local minima. In this note we generalize this result to more general functions of the distances.

Topics: Multidimensional Scaling, SMACOF

3
3.0

Apr 19, 2019
04/19

Apr 19, 2019
by
Jan de Leeuw

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We develop R and C code for Individual Diﬀerence Multidimensional Scaling, both for the INDSCAL and the IDIOSCAL case. In addition to the new SMACOF algorithms to minimize the stress loss function we use expressions for the second derivatives with respect to the group conﬁguration and the individual weights to compute perturbation ellipsoids.

Topics: Multidimensional Scaling, INDSCAL, IDIOSCAL, Stability, Confidence Regions, R Programming, SMACOF

21
21

Apr 15, 2019
04/19

Apr 15, 2019
by
Jan de Leeuw

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In the usual forms of least squares nonlinear principal component analysis observed variables are quantiﬁed or transformed to optimize low-rank approximations. Thus NLPCA is linear PCA on optimally scaled variables. In this note we extend the approach by allowing for optimally scaled components.

Topics: Numerical Analysis, Principal Components, Alternating Least Squares, Block Relaxation,...

3
3.0

Apr 1, 2019
04/19

Apr 1, 2019
by
Jan de Leeuw

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We provide formulas for the convergence rate of majorization (and block relaxation) algorithms with constraints

Topics: Majorization, MM Algorithm, Convergence Rate

6
6.0

Mar 31, 2019
03/19

Mar 31, 2019
by
Jan de Leeuw

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We give an algorithm, with R code, to minimize the multidimensional scaling loss function proposed in Shepard’s 1962 papers. We show the loss function can be justiﬁed by using the classical rearrangement inequality, and we investigated its diﬀerentiability.

Topics: Multidimensional Scaling, Roger Shepard, Rearrangement Inequality, R Programming

11
11

Dec 26, 2018
12/18

Dec 26, 2018
by
Jan de Leeuw

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Using anarchic distances means using a diﬀerent conﬁguration for each dissimilarity. We give the anarchic version of the smacof majorization algorithm, and apply it to additive constants, individual diﬀerences, and scaling of asymmetric dissimilarities.

Topics: Multidimensional Scaling, Majorization, MM Algorithm, R Programming, SMACOF

9
9.0

Jun 19, 2018
06/18

Jun 19, 2018
by
Jan de Leeuw

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We present an R/C implementation of optimal simultaneous diagonalization of several real symmetric matrices using Jacobi plane rotations, with compact triangular storage of symmetric matrices.

Topics: Numerical Analysis, Arrays, Multiway Analysis, Simultaneous Diagonalization, Jacobi Rotations, R...

13
13

May 24, 2018
05/18

May 24, 2018
by
Jan de leeuw

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Earlier papers have shown that stress is diﬀerentiable at local minima if certain conditions on the weights and dissimilarities are satisﬁed. In this note we show the result remains true without these additional conditions.

Topics: Multidimensional Scaling, Stress, Differentiability

4
4.0

May 7, 2018
05/18

May 7, 2018
by
Jan de Leeuw

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The positive orthant method tries to ﬁnd solutions to consistent systems of inequalities, and approximate solutions to inconsistent systems, by maximizing a ﬁt measure based on the sign function and the absolute value function. We concentrate on systems of linear inequalities and develop a convergent majorization algorithm.

Topics: Optimization, Inequalities, Majorization, MM Algorithm, R Programming

23
23

May 4, 2018
05/18

May 4, 2018
by
Jan de Leeuw

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We discuss two diﬀerent even and convex non-negative smooth approximations of the absolute value function and apply them to construct MM algorithms for least absolute deviation regression. Both uniform and sharp quadratic majorizations are constructed. As an example we use the Boston housing data. In our example sharp quadratic majorization is typically 10-20 times as fast as uniform quadratic majorization.

Topics: Optimization, MM Algorithm, R Programming, Majorization

14
14

Apr 9, 2018
04/18

Apr 9, 2018
by
Jan de Leeuw

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The Cauchy-Schwartz majorization of the distance function in SMACOF is replaced by a majorization of the squared distance function. This leads to an interesting SMACOF alternative, which we call SMOCAF.

Topics: Multidimensional Scaling, Majorization, MM Algorithm, R Programming, SMACOF

10
10.0

Feb 24, 2018
02/18

Feb 24, 2018
by
Jan de Leeuw, C. Roger Nance

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This paper is a non-technical and mostly graphical introduction to Homogeneity Analysis, also known as Multiple Correspondence Analysis. It is meant as an explanation and justiﬁcation of a non-standard application of Correspondence Analysis to an example from archeology.

Topics: Multivariate Analysis, Homogeneity Analysis, Gifi, Multiple Correspondence Analysis, Archeology

12
12

Oct 10, 2017
10/17

Oct 10, 2017
by
Jan de Leeuw

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The smacof algorithm for (metric, Euclidean, least squares) multidimensional scaling is rewritten so that all computation is done in C, with only the data management, memory allocation, iteration counting, and I/O handled by R. All symmetric matrices use compact, lower triangular, column-wise storage. Second derivatives of the loss function are provided, but non-metric scaling, individual diﬀerences, and constraints still have to be added.

Topics: Multidimensional Scaling, SMACOF, Majorization, MM Algorithm, R Programming

9
9.0

Aug 31, 2017
08/17

Aug 31, 2017
by
Jan de Leeuw

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We give R, C, and R->C code to access lineary stored multidimensional arrays and compactly stored multidimensional super-symmetric arrays.

Topics: Numerical Analysis, Multidimensional Arrays, R Programming

2
2.0

Aug 6, 2017
08/17

Aug 6, 2017
by
Jan de Leeuw

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We deﬁne fDistances, which generalize Euclidean distances, squared distances, and log distances. The least squares loss function to ﬁt fDistances to dissimilarity data is fStress. We give formulas and R/C code to compute partial derivatives of orders one to four of fStress, relying heavily on the use of Faà di Bruno’s chain rule formula for higher derivatives.

Topics: Multidimensional Scaling, SMACOF, R Programming

8
8.0

Jul 11, 2017
07/17

Jul 11, 2017
by
Jan de Leeuw

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The Jacobi method for computing eigenvalues and eigenvectors of a symmetric matrix is implemented in C using column-wise compact storage of the lower triangle. The complied C code can be loaded into R using the .C() interface. We compare the C implementation with an earlier version in pure R, and with the built-in eigen function in R.

Topics: Numerical Analysis, Linear Algebra, Jacobi Method, Eigenvalues, R Programming

16
16

Jun 3, 2017
06/17

Jun 3, 2017
by
Jan de Leeuw

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We give a majorization algorithm for weighted least squares low-rank matrix approximation, a.k.a. principal component analysis. The loss function has one non-negative weight for each squared residual. A quadratic programming method is used to compute optimal rank-one weights for the majorization scheme.

Topics: Numerical Analysis, Linear Algebra, Majorization, MM Algorithm, R Programming

19
19

May 26, 2017
05/17

May 26, 2017
by
Jan de Leeuw

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In many situations in numerical analysis least squares loss functions with diagonal weight matrices are much easier to minimize than least square loss functions with full positive semi-deﬁnite weight matrices. We use majorization to replace problems with a full weight matrix by a sequence of diagonal weight matrix problems. Diagonal weights which optimally approximate the full weights are computed using a simple semi-deﬁnite programming procedure.

Topics: Optimization, Weighted Least Squares, Majorization, MM Algorithm, R Programming

3
3.0

May 21, 2017
05/17

May 21, 2017
by
Jan de Leeuw

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We give necessary and suﬃcient conditions for solvability of A j = XW j X 0 , with the A j are m given positive semi-deﬁnite matrices of order n. The solution X is n × p and the m solutions W j are required to be diagonal, positive semi-deﬁnite, and adding up to the identity. We do not require that p ≤ n.

Topics: Numerical Analysis, Arrays, Multiway Analysis, Simultaneous Diagonalization

6
6.0

May 5, 2017
05/17

May 5, 2017
by
Jan de Leeuw

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A general form of linear factor analysis is deﬁned, and presented as a method to factor a data matrix, similar in many respects to principal component analysis. We discuss necessary and suﬃcient conditions for solvability of the factor analysis equations and give a constructive method to compute all solutions. A follow up paper will present the corresponding algorithm.

Topics: Multivariate Analysis, Factor Analysis

17
17

Apr 17, 2017
04/17

Apr 17, 2017
by
Jan de Leeuw

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A brief introduction to spline functions and B-splines, and speciﬁcally to monotone spline functions – with code in R and C and with some applications.

Topics: Numerical Analysis, B Splines, Monotone Splines, R Programming

3
3.0

Apr 1, 2017
04/17

Apr 1, 2017
by
Jan de Leeuw

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A C implementation of Kruskal’s up-and-down-blocks monotone regression algorithm for use with .C() is extended to include the three classic ways of handling ties. It is then compared with other implementations.

Topics: Optimization, Isotone Regression, Monotone Regression, R Programming

9
9.0

Mar 30, 2017
03/17

Mar 30, 2017
by
Jan de Leeuw

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A C implementation of Kruskal’s up-and-down-blocks monotone regression algorithm for use with .C(), and a comparison with other implementations.

Topics: Optimization, Isotone Regression, Monotone Regression, R Programming

6
6.0

Mar 22, 2017
03/17

Mar 22, 2017
by
Jan de Leeuw

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We use the system qsort to write a routine that produces both the sort an the order of a vector of doubles.

Topics: Numerical Analysis, Sorting, R Programming

4
4.0

Jan 16, 2017
01/17

Jan 16, 2017
by
Jan de Leeuw

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We give an algorithm, with R code, to minimize the multidimensional scaling stress loss function under the condition that some or all of the ﬁtted distances are between given positive upper and lower bounds. This paper combines theory, algorithms, code, and results of De Leeuw (2017b) and De Leeuw (2017a).

Topics: Multidimensional Scaling, R Programming, SMACOF, MM Algorithm, Majorization

4
4.0

Jan 13, 2017
01/17

Jan 13, 2017
by
Jan de Leeuw

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We give an algorithm, with R code, to minimize the multidimensional scaling stress loss function under the condition that some or all of the ﬁtted distances are smaller than given upper bounds.

Topics: Multidimensional Scaling, R Programming, Majorization, MM Algorithm, SMACOF

10
10.0

Jan 13, 2017
01/17

Jan 13, 2017
by
Jan de Leeuw

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We give an algorithm, with R code, to minimize the multidimensional scaling stress loss function under the condition that some or all of the ﬁtted distances are larger than given positive lower bounds. This paper is a companion to De Leeuw (2017). We also give some interesting majorization theory.

Topics: Multidimensional Scaling, R Programming, Majorization, MM Algorithm, SMACOF

11
11

Jan 3, 2017
01/17

Jan 3, 2017
by
Jan de Leeuw

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We give a quick and dirty, but reasonably safe, algorithm for the minimization of a convex quadratic function under convex quadratic constraints. The algorithm minimizes the Lagrangian dual by using a safeguarded Newton method with non-negativity constraints.

Topics: Optimization, Quadratic Programming, Quadratic Constraints, R Programming

4
4.0

Dec 20, 2016
12/16

Dec 20, 2016
by
Jan de Leeuw

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We discuss the problem of ﬁnding an approximate solution to an overdetermined system of linear inequalities, or an exact solution if the system is consistent. Theory and R code is provided for fouractive set methods for non-negatively constrained least squares, one uses alternating least squares, and one uses a nonsmooth Newton method.

Topics: Optimization, Linear Inequalities, R Programming, Alternating Least Squares

2
2.0

Dec 14, 2016
12/16

Dec 14, 2016
by
Jan de Leeuw

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We construct piecewise quadratic majorizers for minimax problems. This is appled to ﬁnding roots of cubics. An application to a Chebyshev versions of MDS loss is also outlined.

Topics: Optimization, Minimax, Majorization, MM Algorithm, R Programming

7
7.0

Dec 11, 2016
12/16

Dec 11, 2016
by
Jan de Leeuw

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We illustrate uniform quadratic majorization, sharp quadratic majorization, and sublevel quadratic majorization using the example of a univariate cubic.

Topics: Optimization, Majorization, MM Algorithm, R Programming

6
6.0

Dec 2, 2016
12/16

Dec 2, 2016
by
Jan de Leeuw

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We study the convergence rate of the ELEGANT algorithm for squared distance scaling by using both observed convergence rates and an analytical expression for the derivative of the algorithmic map.

Topics: Multidimensional Scaling, Squared Distance Scaling, Convergence Rate, R Programming

10
10.0

Nov 23, 2016
11/16

Nov 23, 2016
by
Jan de Leeuw

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In De Leeuw (2008) we studied the derivatives of the least squares rank p approximation in the case of general rectangular matrices. We modify these results for the symmetric positive semi-deﬁnite case, using basically the same derivation. We apply the formulas to compute an expression for the convergence rate of Thomson’s iterative principal component algorithm for factor analysis.

Topics: Numerical Analysis, Low Rank Approximation, Convergence Rate, R Programming

15
15

Nov 10, 2016
11/16

Nov 10, 2016
by
Jan de Leeuw, Patrick Groenen, Raoul Pietersz

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We reproduce the 1975 derivation of the alternating least squares algorithm for squared distance scaling, from an internal report that got lost in the folds of time. In addition, we present a derivation and a substantial speed improvement based on majorization.

Topics: Multidimensional Scaling, Squared Distance Scaling, Majorization, MM Algorithm, R Programming,...

6
6.0

Nov 4, 2016
11/16

Nov 4, 2016
by
Jan de Leeuw

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This short note shows that all block relaxation algorithms can be formulated as majorization algorithms. The result is mostly a curiosity, without any obvious practical applications

Topics: Majorization, Block Relaxation, Optimization, MM Algorithm

12
12

Jun 29, 2016
06/16

Jun 29, 2016
by
Jan de Leeuw

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A low-dimensional multidimensional scaling example is used to illustrate properties of the stress loss function and of diﬀerent iteration methods.

Topics: Multidimensional Scaling, Stress, SMACOF, Majorization, MM Algorithm, Newton Method, R Programming,...

8
8.0

Jun 8, 2016
06/16

Jun 8, 2016
by
Jan de Leeuw

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In Multidimensional Scaling we sometimes ﬁnd that stress does not decrease if we increase dimensionality. This is explained in this note by using the Gower rank. Some examples with small Gower rank are analyzed.

Topics: Multidimensional Scaling, SMACOF, Full-dimensional Scaling

6
6.0

Mar 23, 2016
03/16

Mar 23, 2016
by
Jan de Leeuw

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This is R/C code for a sequence of principal pivot transforms of a matrix, with applications to least squares, inversion, and determinants.

Topics: Numerical Analysis, Linear Algebra, Principal Pivots, R Programming

15
15

Mar 21, 2016
03/16

Mar 21, 2016
by
Jan de Leeuw

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We present R/C code for rank-revealing QR decomposition with application to generalized inverse, linear equality systems, least squares solutions, and null spaces.

Topics: Numerical Analysis, Linear Algebra, R Programming

11
11

Mar 14, 2016
03/16

Mar 14, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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rStress is the weighted sum-of-squares of the differences between dissimilarities and r-th powers of squared distances. We derive a majorization algorithm for the multidimensional scaling loss function rStress, with r small.

Topics: Multidimensional Scaling, rStress, Majorization, MM Algorithm, R Programming

29
29

Mar 7, 2016
03/16

Mar 7, 2016
by
Jan de Leeuw, Masanao Yajima

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R versions of the array manipulation functions of APL are presented. We do not translate the system functions or other parts of the runtime. Also, the current version has does not have the nested arrays of APL-2.

Topics: APL, R Programming, C Programming

13
13

Feb 23, 2016
02/16

Feb 23, 2016
by
Jan de Leeuw

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In this note we give simple functions, using the .C interface in R, to compute the next permutation or combination in the lexicographic order.

Topics: Numerical Analysis, Permutations, Combinations, R Programming

14
14

Feb 23, 2016
02/16

Feb 23, 2016
by
Jan de Leeuw

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Implementation in R/C of Gram-Schmidt orthogonalization.

Topics: Numerical Analysis, Linear Algebra, R Programming

3
3.0

Feb 15, 2016
02/16

Feb 15, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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We study the properties of stress at points where some distances are zero and the configuration matrix is singular.

Topics: Multidimensional Scaling, Stress, Local Minima

12
12

Feb 8, 2016
02/16

Feb 8, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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rStress is the weighted sum-of-squares of the differences between dissimilarities and r-th powers of squared distances. We investigate if De Leeuw (1984) on the differentiability of stress (i.e. rstress with r = ½) at a local minimum generalizes to other values of r.

Topics: Multidimensional Scaling, rStress, Differentiability

11
11

Jan 25, 2016
01/16

Jan 25, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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This paper discusses full-dimensional scaling, which is multidimensional scaling of n points in n dimensions.

Topics: Multidimensional Scaling, SMACOF, R Programming, Full-dimensional Scaling

12
12

Jan 20, 2016
01/16

Jan 20, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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rStress is the weighted sum-of-squares of the differences between dissimilarities and r-th powers of squared distances. This note gives formulas, code, and applications for the second derivatives of rStress.

Topics: Multidimensional Scaling, rStress, Derivatives, R Programming

13
13

Jan 19, 2016
01/16

Jan 19, 2016
by
Jan de Leeuw

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This is an R wrapper for Cran's Fortran function AMALGM from Applied Statistics. 1980. At the R level it adds the three ways of dealing with ties.

Topics: Optimization, Quadratic Programming, Isotone Regression, Monotone Regression, R Programming

7
7.0

Jan 14, 2016
01/16

Jan 14, 2016
by
Jan de Leeuw, Patrick Groenen, Patrick Mair

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rStress is the weighted sum-of-squares of differences between dissimilarities and r-th power of squared distances. We give a majorization algorithm, and R code, for both r > ½ and r <= ½.

Topics: Multidimensional Scaling, rStress, Majorization, MM Algorithm, R Programming

17
17

Dec 16, 2015
12/15

Dec 16, 2015
by
Jan de Leeuw

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An analysis of the "horseshoe effect" in Multiple Correspondence Analysis. Where do they come from, and are they harmful ?

Topics: Multivariate Analysis, Gifi, R Programming, Correspondence Analysis

18
18

Dec 1, 2015
12/15

Dec 1, 2015
by
Jan de Leeuw

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The Burt matrix collects all bivariate cross tables, and/or covariance matrices, of m variables in a single matrix. Various forms of canonical analysis based on the Burt matrix are discussed.

Topics: Multivariate Analysis, R Programming, Gifi, Correspondence Analysis

2
2.0

Nov 12, 2015
11/15

Nov 12, 2015
by
Jan de Leeuw

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The least squares loss function used in Gifi is rewritten in a slightly different form, with considerable algorithmic implications. Code and examples in R are included.

Topics: Multivariate Analysis, Gifi, R Programming

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14

May 25, 2015
05/15

May 25, 2015
by
Jan de Leeuw

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Suppose we have a dataframe with n observations on m variables, i.e. m vectors in ℝ^n. We want to transform the m variables in such a way that a particular aspect of the correlation matrix of the variables is maximized. An aspect is a real valued function f deﬁned on the space of correlation matrices.

Topics: Multivariate Analysis, Aspects, Gifi, R Programming